Two-loop electroweak threshold corrections to the bottom and top Yukawa couplings

We study the relationship between the MS¯ Yukawa coupling and the pole mass for the bottom and top quarks at the two-loop electroweak order O(α2)O(α2) in the gaugeless limit of the standard model. We also consider the MS¯ to pole mass relationships at this order, which include tadpole contributions to ensure the gauge independence of the MS¯ masses. In order to suppress numerically large tadpole contributions, we propose a redefinition of the running heavy-quark mass in terms of the MS¯ Yukawa coupling. We also present Δr   in the MS¯ scheme at O(α2)O(α2) in the gaugeless limit. As an aside, we also list the exact two-loop expression for the mass counterterms of the bottom and top quarks.